Determine the indefinite integral ∫ln(x2)x3 dx\int \frac{\ln(x^2)}{x^3} \, \text{d}x∫x3ln(x2)dx.
A mechanical piston is subject to a variable resistive force F(x)F(x)F(x), where xxx is the displacement in metres from the start of the stroke. The force, in Newtons, is given by
F(x)=3+2x2+ln(x2)x3,x≥1 F(x) = \frac{3 + 2x^2 + \ln(x^2)}{x^3}, \quad x \ge 1 F(x)=x33+2x2+ln(x2),x≥1The work done by the force as the piston moves from x=1x = 1x=1 to x=2x = 2x=2 is given by ∫12F(x) dx\int_1^2 F(x) \, \text{d}x∫12F(x)dx.
Using the result from part (a), find the exact work done, writing your answer in the form a+lnba + \ln ba+lnb, where aaa and bbb are constants to be determined.
Practise Edexcel A Level Maths Integration with exam-style questions for A Level Maths. 437 questions covering 11.1 Integrating Standard Functions, 11.2 Integrating f(ax + b), 11.3 Using Trigonometric Identities, 11.4 Reverse Chain Rule, 11.5 Integration by Substitution, 11.6 Integration by Parts, 11.7 Partial Fractions, 11.8 Finding Areas, 11.9 The Trapezium Rule, 11.10 Solving Differential Equations, and 11.11 Modelling with Differential Equations, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
